Since the membership condition for \(P̏(X)\)  requires quantification over the open-ended totality \(V̥_0\)  of predicative sets (see [1]}, chapter 2), the totality \(P̏(X)\)  is a proper class. It is immediate to show that \(f \colon A \subseteq B\)  is an embedding, and that the category \(P̏(X)\)  is thin.
